There is beaucoup public land in AZ, so finding a place to hunt isn't hard at all.
The biggest PITA is to make sure you are on the right kind of land - State Trust land is good to go, BLM is if you have a valid hunting license and permit. You'll need to look at the regs (they seem to change every year, so look it up for the most current stuff) and make sure you have the documentation you need.
One thing that his a major headache is that there is no designation out in the desert as to what land you are on, so a GW could give you a ticket if you were inadvertently on the wrong type of land. It is very, very, very rare that you'll ever run into a GW, and even rarer that one would ticket you for such an offense, but keep in mind it is out there.
As far as where to hunt, I had good luck on the NW side of town around the White Tank Mountains area, on over west to Flat Iron Mountain, and then further north up towards Lake Pleasant. Those areas are generally scrub brush, which provided a lot of cover to both dove and quail, and were extremely vast so they had plenty of places to feed and water.
On the Southeast side of town, you have to be more careful because of the SRPMIC and South Moutnain reservations. Hunting on res land without the proper permits is bad ju-ju. There was some pretty good dove hunting near the San Tan range off of 87, but I don't have a whole lot of experience outside of a couple of hunts here and there.
A good website to ask specific questions is
Arizona Shooting.